Pro t-Maximizing Matchmaker

This paper considers a matchmaker game in the Shapley-Shubik(1971) (one-to-one) assignment problem. Each rm proposes how muchit is willing to pay each worker if they are matched. Each worker alsoproposes which salary she is willing to accept from each rm if they arematched. The matchmaker chooses a matching to maximize pro t (thesum of the di¤erence between the o¤ering and asking salaries from eachmatched rm-worker). First, we show that Nash equilibrium may gen-erate ine¢ cient outcomes, but the matchmakers pro t is always zero inevery Nash equilibrium. Second, we show that the sets of stable assign-ments and strong Nash equilibria are equivalent. These results extendto the Kelso-Crawford (1982) many-to-one assignment problem. Inter-estingly, in the one-to-one matching case, our results are closely relatedto the common agency game by Bernheim and Whinston (1986), whilein the many-to-one assignment problem, such relationships break downcompletely....